One of the main differences is that oils are liquid at ambient temperature when compared to fats that are solids (generally) at ambient temperatures.
There are also oils that are composed of and by non-organic substances and processes, including the oil we most often think of as the black liquid we get out of the ground along with its various components. These "mineral oils" did originally come from the organic material of plankton or other small organisms but were converted into mineral oils through eons of pressure and heat.
Usually fats that cross over and are liquid at ambient temperature are referred to as lipids.
Otherwise the two are similar in many ways, particularly that they are very efficient for storing energy both for animal use and processing and use in machines, etc.